U.S. patent number 6,845,164 [Application Number 09/947,755] was granted by the patent office on 2005-01-18 for method and device for separating a mixture of source signals.
This patent grant is currently assigned to Telefonaktiebolaget LM Ericsson (PUBL). Invention is credited to Tony Gustafsson.
United States Patent |
6,845,164 |
Gustafsson |
January 18, 2005 |
Method and device for separating a mixture of source signals
Abstract
Device and method for separating a mixture of source signals to
regain the source signals, the device and method being based on
measured signals, the invention comprises: bringing each measured
signal to a separation structure including an adaptive filter, the
adaptive filter comprising filter coefficients; using a generalized
criterion function for obtaining the filter coefficients, the
generalized criterion function comprising cross correlation
functions and a weighting matrix, the cross correlation functions
being dependent on the filter coefficients; estimating the filter
coefficients, the resulting estimates of the filter coefficients
corresponding to a minimum value of the generalized criterion
function; and updating the adaptive filter with the filter
coefficients.
Inventors: |
Gustafsson; Tony (Molndal,
SE) |
Assignee: |
Telefonaktiebolaget LM Ericsson
(PUBL) (Stockholm, SE)
|
Family
ID: |
20414782 |
Appl.
No.: |
09/947,755 |
Filed: |
September 7, 2001 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
Issue Date |
|
|
PCTSE0000451 |
Mar 7, 2000 |
|
|
|
|
Current U.S.
Class: |
381/94.1; 381/66;
702/190 |
Current CPC
Class: |
G06K
9/6243 (20130101); G06K 9/6245 (20130101); H03H
21/0012 (20130101) |
Current International
Class: |
H03H
21/00 (20060101); H04B 015/00 () |
Field of
Search: |
;381/94.1,94.2,94.3,94.7,94.9,66 ;708/322 ;706/22
;702/190,191,194,195 ;375/316 |
References Cited
[Referenced By]
U.S. Patent Documents
Foreign Patent Documents
Primary Examiner: Mei; Xu
Attorney, Agent or Firm: Burns, Doane, Swecker & Mathis,
L.L.P.
Parent Case Text
This application is a continuation of International Application No.
PCT/SE00/00451 filed on Mar. 7, 2000.
Claims
What is claimed is:
1. A method for separating a mixture of source signals to regain
the source signals, the method being based on measured signals, the
method comprising: bringing each measured signal to a separation
structure including an adaptive filter, the adaptive filter
comprising filter coefficients; using a generalized criterion
function for obtaining the filter coefficients, the generalized
criterion function comprising cross correlation functions and a
weighting matrix, the cross correlation functions being dependent
on the filter coefficients; said weighting matrix, being an inverse
matrix of a matrix comprising an estimation of a covariant matrix
for a signal; said signal having a spectrum being a product of
estimated spectrum of an incoming source-signal and the determinant
of an estimated transformation function of a mixing filter;
estimating the filter coefficients, the resulting estimates of the
filter coefficients corresponding to a minimum value of the
generalized criterion function; and updating the adaptive filter
with the filter coefficients.
2. The method according to claim 1, wherein said spectrum of the
incoming source-signal is unknown and the weighting matrix
considers an assumed spectrum.
3. The method according to claim 1, wherein the weighting matrix is
dependent on the filter coefficients.
4. The method according to claim 1, wherein the method is
repeatedly performed on the measured signal or on fractions
thereof.
5. The method according to claim 4, wherein the method is
repeatedly performed according to a predetermined updating
frequency.
6. The method according to claim 1, wherein the number of filter
coefficients is predetermined.
7. A device for separating a mixture of source signals to regain
the source signals, the input to the device being based on measured
signals, the device comprising: signaling links for bringing each
measured signal to a separation structure including an adaptive
filter, the adaptive filter comprising filter coefficients; a
generalized criterion function means for obtaining the filter
coefficients, the generalized criterion function means comprising
cross correlation functions and a weighting matrix, the cross
correlation functions being dependent on the filter coefficients;
said weighting matrix, being an inverse matrix of a matrix
comprising an estimation of a covariant matrix for a signal; said
signal having a spectrum being a product of estimated spectrum of
an incoming source-signal and the determinant of an estimated
transformation function of a mixing filter; means for estimating
the filter coefficients, the resulting estimates of the filter
coefficients corresponding to a minimum value output of the
generalized criterion function; and updating means for updating the
adaptive filter with the filter coefficients.
8. The device according to claim 7, wherein the weighting matrix is
dependent on the filter coefficients.
9. The device according to claim 7, wherein the device is arranged
to separate the measured signals or fractions thereof
repeatedly.
10. The device according to claim 9, wherein the device is arranged
to separate the measured signals or fractions thereof according to
a predetermined updating frequency.
11. The device according to claim 7, wherein the number of filter
coefficients is arranged to be predetermined.
Description
TECHNICAL FIELD
The present inventions relates to a metod and device for separating
a mixture of source signals to regain the source signals.
BACKGROUND OF THE INVENTION
In recent time several papers concerning signal separation of
dynamically mixed source signals have been put forward [1-3, 8, 17,
19, 20]. In principle it is possible to separate the sources
exploiting only second order statistics, cf. [8]. The blind signal
separation problem with dynamic/convolutive mixtures is solved in
the frequency domain in several papers presented, cf. [3, 20].
Basically, dynamic source separation in the frequency domain aims
to solve a number of static/instantaneous source separation
problems, one for each frequency bin in question. In order to
obtain the dynamic channel system (mixing matrix), the estimates
corresponding to different frequencies bins, have to be
interpolated. This procedure seems to be a nontrivial task, due to
scaling and permutation indeterminacies [16]. The approach in the
present paper is a "time-domain approach", see [8], which models
the elements of the channel system with Finite Impulse Response
(FIR) filters, thus avoiding this indeterminacies.
A quasi-maximum likelihood method for signal separation by second
order statistics is presented by Pham and Garat in [11]. An
algorithm is presented for static mixtures, i.e. mixing matrices
without delays. Each separated signal s.sub.i i=1, . . . , M is
filtered with a Linear Time Invariant (LTI) filter h.sub.i. The
criterion used is the estimated cross-correlations for these
filtered signals. The optimal choice of the filter h.sub.i,
according to [11], is the filter with frequency response inversely
proportional to the spectral density of the corresponding source
signal. The filters h.sub.i, i=1, . . . , M are thus whitening
filters. However, the spectral densities of the source signals are
usually unknown, and perhaps time varying. One approach is to
estimate these filters as done in the present paper and in the
prediction error method as presented in [1]. Moreover, several
aspects of the algorithm presented in [8] remained open.
SUMMARY OF THE INVENTION
Characterizing features of the present invention, i.e. the method
for separating a mixture of source signals to regain the source
signals; are bringing each measured signal to a separation
structure including an adaptive filter, the adaptive filter
comprising filter coefficients; using a generalised criterion
function for obtaining the filter coefficients, the generalised
criterion function comprising cross correlation functions and a
weighting matrix, the cross correlation functions being dependent
on the filter coefficients; estimating the filter coefficients, the
resulting estimates of the filter coefficients corresponding to a
minimum value of the generalised criterion function; and updating
the adaptive filter with the filter coefficients.
Other characterizing features of the present invention, i.e. the
device for separating a mixture of source signals to regain the
source signals, the input to the device being based on measured
signals, the device comprises: signaling links for bringing each
measured signal to a separation structure including an adaptive
filter, the adaptive filter comprising filter coefficients; a
generalised criterion function means for obtaining the filter
coefficients, the generalised criterion function means comprising
cross correlation functions and a weighting matrix, the cross
correlation functions being dependent on the filter coefficients;
means for estimating the filter coefficients, the resulting
estimates of the filter coefficients corresponding to a minimum
value output of the generalised criterion function; and updating
means for updating the adaptive filter with the filter
coefficients.
Specific fields of application of the present invention include
mobile telephone technology, data communication, hearing aids and
medical measuring equipment, such as ECG. Also included is echo
cancelling which can primarily occur in the telecommunications
field.
BRIEF DESCRIPTION OF THE DRAWINGS
FIGS. 1A-1D show the empirical and true parameter variances of a
preferred embodiment of the present invention compared to a prior
art signal separation algorithms.
FIGS. 2A-2D show the estimated mean value as a function of relative
frequency of a preferred embodiment of the present invention
compared to a prior art signal separation algorithm.
FIGS. 3A-3D show the parameter variances as a function of relative
frequency of a preferred embodiment of the present invention
compared to a prior art signal separation algorithm.
DESCRIPTION OF PREFERRED EMBODIMENTS
In the present invention a signal separation algorithm is derived
and presented. A main result of the analys is an optimal weighting
matrix. The weighting matrix is used to device a practical
algorithm for signal separation of dynamically mixed sources. The
derived algorithm significantly improves the parameter estimates in
cases where the sources have similar color. In addition the
statistical analysis can be used to reveal attainable (asymptotic)
parameter variance given a number of known parameters.
The basis for the source signals, in the present paper, are M
mutually uncorrelated white sequences. These white sequences are
termed source generating signals and denoted by .xi..sub.k (n)
where k=1, . . . , M. The source generating signals are convolved
with linear time-invariant filters G.sub.k (q)/F.sub.k (q) and the
outputs are, ##EQU1##
referred to as the source signals and where q and T is the time
shift operator and matrix transpose, respectively. The following
assumptions are introduced
A1. The generating signal .xi.(n) is a realization of a stationary,
white zero-mean Gaussian process:
A2. The elements of K(q) are filters which are asymptotically
stable and have minimum phase.
Condition A1 is somewhat restrictive because of the Gaussian
assumption. However, it appears to be very difficult to evaluate
some of the involved statistical expectations unless the Gaussian
assumption is invoked.
The source signals x(n) are unmeasurable and inputs to a system,
referred to as the channel system. The channel system produces M
outputs collected in a vector y(n)
which are measurable and referred to as the observables. In the
present paper the channel system, B(q), given in ##EQU2##
where B.sub.ij (q), ij=1, . . . M are FIR filters. The objective is
to extract the source signals from the observables. The extraction
can be accomplished by means of all adaptive separation structure,
cf. [8]. The inputs to the separation structure are the observable
signals. The output from the separation structure, s.sub.1 (n), . .
. ,s.sub.M (n), depend on the adaptive filters, D.sub.ij
(q,.theta.), i,j=1, . . . M, and can be written as
where .theta. is a parameter vector containing the filter
coefficients of the adaptive filters. That is the parameter vector
is .theta.=[d.sub.11.sup.T. . . d.sub.MM.sup.T ].sup.T where
d.sub.ij, i,j=1, . . . M are vectors containing the coefficients of
D.sub.ij (q, .theta.), i,j=1, . . . M, respectively. Note, that
unlike B(q) the separation matrix D(q, .theta.) does not contain a
fixed diagonal, cf. [6, 13].
Most of the expressions and calculations in the present paper will
be derived for the two-input two-output (TITO) case, M=2. The main
reason for using the TITO case is that it has been shown to be
parameter identifiable under a set of conditions, cf. [8]. However,
the analysis in the current paper is applicable on the more general
multiple-input multiple-output (MIMO) case, assuming that problem
to be parameter identifiable as well.
Assuming that N samples of y.sub.1 (n) and y.sub.2 (n) are
available, the criterion function proposed in [8], reads as
##EQU3##
where ##EQU4##
To emphasize the dependence on .theta. equation (2.6) can be
rewritten as ##EQU5##
where the notation d.sub.12 (i) denotes the i:th coefficient of the
filter D.sub.12 (q).
For notational simplicity, introduce the following vector as
where the subscript N indicates that the estimated
cross-covariances are based on N samples. Furthermore, introduce a
positive definite weighting matrix W(.theta.) which possibly
depends on .theta. too. Thus, the criterion, in equation (2.6), can
be generalized as ##EQU6##
which will be investigated. Note, the studied estimator is closely
related to the type of non-linear regressions studied in [15]. The
estimate of the parameters of interest are obtained as ##EQU7##
Although the signal separation based on the criterion (2.6) has
been demonstrated to perform well in practice, see for example
[12], there are a couple of open problems in the contribution [8];
1. It would be interesting to find the asymptotic distribution of
the estimate of .theta..sub.N. Especially, an expression for the
asymptotic covariance matrix is of interest. One reason for this
interest, is that the user can investigate the performance for
various mixing structures, without performing simulations.
Potentially, further insight could be gained into what kind of
mixtures that are difficult to separate. The asymptotic covariance
matrix would also allow the user to compare the performance with
the Cramer-Rao Lower bound (CRB), primarily to investigate how far
from the optimal performance of the prediction error method the
investigated method is. An investigation of the CRB for the MIMO
scenario can be found in [14]. 2. How should the weighting matrix
W(.theta.) be chosen for the best possible (asymptotic) accuracy?
Given the best possible weighting, and the asymptotic distribution,
one can further investigate in which scenarios it is worthwhile
applying a weighting W(.theta.).noteq.I, where I denotes the
identity matrix.
The purpose of the present contribution is to:
Find the asymptotic distribution of the estimate of
.theta..sub.N.
Find the weighting matrix W(.theta.) that optimizes the asymptotic
accuracy.
Study an implementation of the optimal weighting scheme.
In addition to A1 and A2 the following assumptions are considered
to hold throughout the description: A3. Assume that the conditions
C3-C6 in [8] are fulfilled, so that the the studied TITO system is
parameter identifiable. A4. The (minimal) value of U is defined as
in Proposition 5 in [8]. A5.
.parallel..theta..parallel.<.infin., i.e. .theta..sub.0 is an
interior point of a compact set D.sub.M. Here, .theta..sub.0
contains the true parameters.
This section deals with the statistical analysis and it will begin
with consistency. The asymptotic properties (as N-.infin.) of the
estimate of .theta..sub.N (.theta.^.sub.N) is established
in the following. However, first some preliminary observations are
made. In [8] it was shown that
1. As N.fwdarw..infin., R,.sub.1,.sub.2 (k,
.theta.).fwdarw.R,.sub.1,.sub.2 (k, .theta.) with probability one
(w.p.1). Thus
where ##EQU8##
The convergence in (3.1) is uniform in a set D.sub.M, where .theta.
is a member ##EQU9##
Furthermore, since the applied separation structure is of finite
impulse response (FIR) type, the gradient is bounded ##EQU10##
for N larger than some N.sub.0 <.infin.. In equation (3.4), C is
some constant, C<.infin., and n.theta. denotes the dimension of
.theta.. The above discussion, together with the identifiability
analysis [8] then shows the following result:
Result 1 As N.fwdarw..infin.,
Having established (strong) consistency, the asymptotic
distribution of .theta.^.sub.N is considered. Since the
.theta.^.sub.N minimizes the criterion V.sub.N (.theta.),
V.sub.N (.theta.^.sub.N)=0, where V.sub.N denotes the
gradient of V.sub.N. By the mean value theorem,
where .theta..sub..xi. is on a line between .theta.^.sub.N
and .theta..sub.0. Note, since .theta.^.sub.N is
consistent, the .theta.^.sub.N -.theta..sub.0 and
consequently, .theta..sub..xi..fwdarw..theta..sub.0, as
N.fwdarw..infin..
Next, investigate the gradient evaluated at .theta..sub.0 (for
notational simplicity, let W(.theta.)=W) ##EQU11##
where ##EQU12##
Note, evaluation of G is straightforward, see for example [8]. The
introduced approximation does not affect the asymptotics, since the
approximation error goes to zero at a faster rate than does the
estimate of r.sub.N (.theta..sub.0) (r^.sub.N
(.theta..sub.0)). Furthermore, since r.sub.N (.theta..sub.0)=0, the
asymptotic distribution of V.sub.N '(.theta.)
is identical to the asymptotic distribution of G.sup.T
Wr^.sub.N (.theta..sub.0), where ##EQU13##
Applying for example Lemma B.3 in [15], and using the fact that
both s.sub.1 (n:.theta..sub.0) and s.sub.2 (n;.theta..sub.0)
stationary ARMA processes, one can show that (√N)G.sup.T
Wr^.sub.N (.theta..sub.0) converges in distribution to a
Gaussian random vector, i.e.
where ##EQU14##
This means that the gradient vector, is asymptotically normally
distributed, with zero-mean and with a covariance matrix M.
Before presenting the main result of the current paper, the
convergence of the Hessian matrix V".sub.N must be investigated.
Assuming that the limit exists, define ##EQU15##
To establish the convergence of V".sub.N (.theta..sub..xi.), the
following (standard) inequality is applied ##EQU16##
where .parallel. .parallel..sub.F denotes the Frobenius norm. Due
to the FIR separation structure, the second order derivatives are
continuous. Moreover, since .theta..sub..xi. converges w.p.1 to
.theta..sub.0, the first term converges to zero w.p.1. The second
term converges also to zero w.p.1. This can be shown using a
similar methodology that was used to show (3.3). Note also that
since the third order derivatives are bounded, the convergence is
uniform in .theta..
It is, now, straightforward to see that the limiting Hessian
V.sup.- " can be written as ##EQU17##
Thus, for large N,
assuming the inverses exists (generically guaranteed by the
identifiability conditions in A3. Here all approximation errors
that goes to zero faster than O (1/√(N)) have been neglected.
Finally, the following result can be stated.
Consider the signal separation method based on second order
statistics, where .theta.^.sub.N is obtained from (2.10).
Then the normalized estimation error, √(N)(.theta.^.sub.N
-.theta..sub.N), has a limiting zero-mean Gaussian distribution
√N(.theta..sub.N -.theta..sub.0).epsilon.AsN(0, P),
where
Obviously, the matrix M plays a central role, and it is of interest
to find a more explicit expression. For simplicity we consider only
the case when the generating signals are zero-mean, Gaussian and
white (as stated in Assumption A1). It seems to be difficult to
find explicit expressions for the non-Gaussian case. Note also that
this is really the place where the normality assumption in A1 is
crucial. For example, the asymptotic normality of
√(N)r^.sub.N (.theta..sub.0) holds under weaker
assumptions.
Theorem 6.4.1 in [5] indicates precisely how the components of M
can be computed. These elements are actually rather easy to
compute, as the following will demonstrate. Let ##EQU18##
Furthermore, introduce the following Z-transforms ##EQU19##
Then it follows that ##EQU20##
Thus, the .beta..sub..tau. 's are the covariances of an ARMA
process with power spectrum ##EQU21##
Computation of ARMA covariances is a standard topic, and simple and
efficient algorithms for doing this exists, see for example [15,
Complement C7.7]. Given .beta..sub..tau. for .tau.=0 . . . , 2U,
the weighting matrix can, hence, be constructed as ##EQU22##
Thus, in the present problem formulation the separated signals are
distorted with the determinant of the channel system the channel
system determinant equals det {B(z)}, and one may define the
reconstructed signals as ##EQU23##
as long as det{D(q, .theta..sub.0)} is minimum phase.
To complete our discussion, it is also pointed out how the matrix G
can be computed. The elements of G are all obtained in the
following manner. Using Equation (2.8). it follows that
##EQU24##
which are straightforward to compute.
Next, consider the problem of choosing W. Our findings are
collected in the following result. The asymptotic accuracy of
.theta.^.sub.N, obtained as the minimizing argument of the
criterion (2.10), is optimized if
For this choice of weighting,
The accuracy is optimized in the sense that P(W.sub.0)-P(W) is
positive semi-definite for all positive definite weighting matrices
W.
The proof follows from well-known matrix optimization results, see
for example [9, Appendix 2].
The result could have been derived directly from the ABC theory in
[15, Complement C4.4]. However, the result above itself is a useful
result motivating the presented analysis.
Before considering the actual implementation of the optimal
weighting strategy, let us make a note on the selection of U. This
parameter is a user-defined quantity, and it would be interesting
to gain some insight into how it should be chosen. Note, Assumption
A3 states a lower bound for U with respect to identifiability. The
following result may be useful.
Assume that the optimal weighting W.sub.0 is applied in the
criterion (2.10). Let P.sub.U (W) denote the asymptotic covariance
for this case. Then
The proof follows immediately from the calculations in [15,
Complement C4.4]. Note that, when the optimal weighting, W.sub.0 is
applied the matrices {P.sub.U (W.sub.0)} forms a non-increasing
sequence. However, in practice one must be aware that a too large
value of U in fact may deteriorate the performance. This phenomenon
may be explained by that a large value of U means that a larger
value of N is required in order for the asymptotic results to be
valid.
In the present section a comparison of signal separation based on
an algorithm within the scope of the present invention and the
algorithm in [8] will be made. The purpose for the comparison is to
show the contribution of the present invention. Put differently,
does the weighting lead to a significant decrease of the parameter
variance? In all of our simulations, U=6. Furthermore, the term
relative frequency is used in several figures. Here relative
frequency corresponds to f.sub.rel =2F/F.sub.S where F.sub.S is
the
Here the channel system is defined by B.sub.12 (q)=0.3+0.1q.sup.-1
and B.sub.21 (q)=0.1+0.7q.sup.-1. The source signal x.sub.1 (n) is
an AR(2) process with poles at radius 0.8 and angles .pi./4. The
second source signal is, also, an AR(2) process. However, the poles
are moved by adjusting the angles in the interval [0,.pi./2], while
keeping the radius constant at 0.8. At each angle 200 realizations
have been generated and processed by the channel system and
separation structure. That is to say, for each angle the resulting
parameter estimates have been averaged. Finally, each realization
consists of 4000 samples.
In FIGS. 1A-1D the empirical and true parameter variances are
depicted. First, note the good agreement between the empirical and
theoretical variances. Second, observe that the proposed weighting
strategy for most angles gives rise to a significant variance
reduction. In FIGS. 1A-1D, the parameter variances as a function of
relative frequency. "*" denotes empirical variance of the prior art
signal separation algorithm; "+" denotes empirical variance of the
proposed weighting strategy. The solid line is the true asymptotic
variance of the unweighted algorithm, and the dashed line is the
true asymptotic variance for the optimally weighted algorithm. The
dotted line is the CRB.
Apparently, it gets more difficult to estimate the channel
parameters when the source colors are similar. Therefore, in FIGS.
2A-2D and FIGS. 3A-3D a more careful examination of the parameter
accuracy is presented. The angles of the poles are in the interval
[40.degree., 50.degree.]. In FIGS. 2A-2D, the estimated mean value
is depicted as a function of relative frequency. "*" denotes
empirical mean value of the prior art signal separation algorithm;
"+" denotes empirical mean value of the proposed weighting
strategy. The solid lines correspond to the true parameter values.
Note that the optimally weighted algorithm gets biased, although
less biased than a prior art signal separation algorithm. This bias
is probably an effect of the fact that the channel estimates of the
unweighted algorithm are rather inaccurate, making the weighting
matrix inaccurate as well. In FIGS. 3A-3D, the parameter variances
is depicted as a function of relative frequency; "*" denotes
empirical variance of the prior art signal separation algorithm
[8]; "+" denotes empirical variance of the proposed weighting
strategy. The solid line is the true asymptotic variance of the
unweighted algorithm, and the dashed line is the true asymptotic
variance for the optimally weighted algorithm. The dotted line is
the CRB. As is indicated by the FIGS. 1A-1D, FIGS. 2A-2D, and FIGS.
3A-3D, the present invention increases the quality of the signal
separation.
Further details of the present invention are that the method,
according the device for separating signals, is repeatedly
performed on the measured signal or on fractions thereof. Also, the
method may be repeatedly performed according to a predetermined
updating frequency. It should be noted that the predetermined
updating frequency may not be constant. Further, the number of
filter coefficients is predetermined. Finally, the number of filter
coefficients is arranged to be predetermined in the above
embodiment.
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* * * * *